In the introductory tutorial in this series we considered a harmonic forcing function **F _{0}cos(ωt)** applied to a translational spring and mass system with one degree of freedom having mass

**m**, spring constant

**k**and a dashpot providing viscous damping with damping constant

**c**shown in Figure 1 below with the spring extended by distance

**x**.

From the dynamic* free body diagram we obtained the following equation of motion in terms of displacement **x** and time **t**:

**sum of forces produces acceleration of mass m, not static equilibrium*

We now find a function x(t) that is a general solution to this equation.

*the maths tutorial in this series provides an outline of solutions to homogeneous and non-homogeneous second-order ordinary differential equations*

Recall that the **general solution** to a second-order non-homogeneous ordinary linear differential equation is:

**the general solution for the complementary homogeneous equation**

**+**

**+**

**the particular solution for the non-homogeneous equation**

For forced vibrations with damping the first part of the solution is the general solution to the homogeneous differential equation of motion for ** damped free vibrations** derived in a previous tutorial as follows:

where **C _{1}** and

**C**are constants,

_{2}**ω**is the natural frequency of the spring and mass system

_{n}and

**ζ = c/2ω**is the damping ratio

_{n}.mWe also showed that all particular solutions of equation (2) (where ζ > 1, ζ = 1 and ζ < 1) have the characteristic that x(t) → 0 as time increases and can thus be considered a **transient condition** for forced vibrations with damping.

In this tutorial we are interested in the **steady state condition** which is defined by the particular solution to equation (1) as this has much greater practical significance. Hence we will only consider the particular solution.

#### Find the particular solution to equation (1)

We know that the particular solution to equation (1) will be a harmonic function in **ωt** where **ω** is the angular frequency of the forcing function and will have the form:

** x _{p} = A.cos(ωt) + B.sin(ωt)** where

**A**and

**B**are unknown values of displacement

**x(t)**.

This expression can also be stated in the form:

**x _{p} = X.cos(ωt - φ) ------ (3)**

where **X** is an unknown value of **x(t)** and **φ** is an unknown value of **phase angle**.

We now proceed to solve equation (3) for unknowns **X** and **φ**.

Firstly, substitute **x _{p}**,

**dx**and

_{p}/dt**d**derived from equation (3) into equation (1) to give:

^{2}x_{p}/dt^{2}Now divide by **k** to give:

**see a previous tutorial for the derivation of damping ratio ζ *

We now create two equations from equation (4) to provide solutions for the two unknowns, **X** and **φ** using the following trig identities:

(i) **cos(ωt - φ) = cos(ωt).cos(φ) + sin(ωt).sin(φ)**

(ii) **sin(ωt - φ) = sin(ωt).cos(φ) - cos(ωt).sin(φ)**

**X.cos(ωt).[A.cos(φ) + B.sin(φ)] + X.sin(ωt).[A.sin(φ) - B.cos(φ)] = (F _{0}/k).cos(ωt) ------- (5)**

By equivalence of coefficients in equation (5) for **cos(ωt)** and **sin(ωt)** terms gives the following:

(i) **X.[A.cos(φ) + B.sin(φ)] = (F _{0}/k) ---- (6)**

(ii) **X.[A.sin(φ) - B.cos(φ)] = 0 ---- (7)**

Solving simultaneous equations (6) and (7) gives the following solutions for **X** and **φ** :

Substituting back for **A** and **B** gives:

which completes the solution for equation **x _{p} = X.cos(ωt - φ)** representing the

**steady state**condition of a damped forced vibration with forcing function

**F**where:

_{0}cos(ωt).**X** is the **amplitude** of displacement and **φ** the **phase angle** of displacement relative to the harmonic forcing function.

#### Characteristics of steady state response

*Note: for practical application we express the equation for steady state response as*:

**x(t) = X.cos(ωt - φ)**

In the analysis below we use the following important parameters:

**frequency ratio = ω/ω**which is the ratio of the frequency of the harmonic forcing function and the natural frequency of the spring and mass system._{n}**damping ratio ζ = c/2ω**which is a dimensionless factor expressing the degree of damping in the system._{n}.m**static amplitude δ = (F**which expresses the static displacement of mass_{0}/k)**m**subject to force**F**in a spring and mass system with spring constant_{0}**k**.**amplification factor (X/δ)**which expresses the amplification effected by the forcing function relative to the static displacement.

**Examples**

The following two examples plot steady state response of a damped spring and mass system to a harmonic forcing function **F _{0}cos(ωt)**. Damping ratio of the system

**ζ = 0.1**.

Figures 2 below plots the steady state response for a system where frequency ratio **ω/ω _{n} < 1 **. The amplification factor

**(X/δ) ≅ 1.75.**. Response

**lags**the forcing function.

Figures 3 below plots the steady state response for a system where frequency ratio **ω/ω _{n} > 1 **. The amplification factor

**(X/δ) ≅ 1.21**. Response

**leads**the forcing function.

**General characteristics of steady state response**

General characteristics of steady state response over a wide range of system parameters are illustrated in Figures 4 and 5 below.

Figure 4 below plots **amplification factor (X/δ) ** against **frequency ratio ω/ω _{n}** for three values of

**damping ratio ζ**.

Note the following from Figure 4:

- The amplification factor peaks at a frequency ratio dependent on
**ζ**. The peak amplification factor occurs closer to frequency ratio = 1 as**ζ**tends to zero, which is the**resonant condition**for undamped systems derived in the previous tutorial. - For all values of
**ζ**the amplification factor tends to zero as frequency ratio increases beyond the peak value. - The frequency ratios used to compute the plots in Figures 2 and 3 above are indicated on the plot line for
**ζ = 0.1**.

Figure 5 below shows the variation of **phase angle φ** (units in radians) with **frequency ratio** for three values of **damping ratio ζ**.

Note the following from Figure 5:

- At frequency ratios < 1 displacement x(t)
**lags**the forcing function. Phase angles increase from zero to π/2 as the frequency ratio increases to 1. - At frequency ratios > 1 displacement x(t)
**leads**the forcing function. Phase angles increase from π/2 to π as the frequency ratio increases. - The frequency ratios used to compute the plots in Figures 2 and 3 above are indicated on the plot line for
**ζ = 0.1**. - At frequency ratio = 1 there is an unstable condition where the phase angle transitions from π/2 lagging to π/2 leading.

Figures 6 and 7 below illustrate the transformation of phase angle from lagging to leading where ω/ω_{n} is respectively fractionally less than and fractionally grater than 1.

This tutorial concludes the series on mechanical vibrations at present.

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